EXERCISE 9.1
1.
BASIC INTEGRATION FORMULAS
− − − 6
2
4 +5 3
6
=
2
4
3
2
+5 +
=
3.
+
+
5.
3 2
= =
+
− − − − − − − 2 2 +4
3
2
= =
2+
2
4
+
3
2
4
=
+
3
2
3
=2 +4
− − ( 2 +2 +4)( (
1
9.
2)
2)
2
= (
=
=
2
=
1)
=
8
= Factor, (x-c), c = 2 P(c) = 0 – the (x-c ) is the factor P(c) = 0 2 1 0 0 -8 2 4 8 1240
− − − − − (
−− 3
7.
+ 2 + 4)
3
3
+
+
2 2 2
+4 +
+
+
− − − − 4
= =
2
3
+
2
2
2
+
2 3
+
+
1
+
+
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EXERCISE 9.2
INTEGRATION BY SUBSTITUTION
− − − − − − − − − − − − − 2
1.
3
Let u = 2 - 3x
=
=
3
1 2
=
(
3
1
=
)
1 2
3
3
1 2 2
=
3
+
3
+
=
2
(2
3
1)4
Let u = 2
3
1
3.
2
=6
6
2
=
=
=
=
2
=
(
1 6 1
3
(2
4
(
)( 4
6
1)4
)
3
− − −− −− − − − (2 +3)
5.
2 +3
2
Let u =
+4
+3 +4
=2 +3
= (2 + 3)
= =
+
+
=
+
+
2
7.
( 3 1)4 3
Let u =
= =
=
2
3 4
= =
= 3
2
=
3
1
1
4
3 1
3
3
3
+
3
+
9
(
)
+
)
5
6 5
+
5
=
=
30
(
+
)
+
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EXERCISE 9.2
INTEGRATION BY SUBSTITUTION
− − − − − − − − − − − − − − − − − − − − 9.
2
Let u =
=
1
=
1
=
( )
2
1
=
2
2
=
1
=
+
1
+
=
11.
1
=
Let u = = (
1
1
1
)
1
=
1
1
Let v =
1
=
=
1
1
=
+
=
= ln ln |
= ln (1 =
1
;
1|
=
ln |
)
+
| +
− − − − − − cos 4 sin
13.
Let u = cos =
=
sin
sin
4
=
4
=-
5
=
+
5
+
=
1+2sin3
15.
3
Let u = 3 =3
=
3
1+2sin
= =
1
(
3
)
1+2sin
3
Let v =1 =1 + 2 s i n ;
=2
= =
2
1+2sin3
1 3
1 2
[
=
3
( )] 2
3
=
=
1 2 2 6
( +
3
+
)
+
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EXERCISE 9.2
INTEGRATION BY SUBSTITUTION
2
17.
`
+
Let u =
+
sin
=
;
2
=
= =
1
+
=
+
3 2 +14 +14
21.
+4
( )
=
( )
=
+
( )
* using synthetic division -4 3 14 13 -12 -8 3 2 5 - R(x)
=3 +2
+4=
( )
= (3 + 2)
19.
3
Let u =
3
2
=3 1 2
=
(
3
2
3 ;
)
3
=
1 3
[
2 2 3
]+
3
3
=
2
3
5
+
+4
For the second integral : =
+4
= (3 + 2) =[ =
3 2 2
;
+
=
+5
+2 +5
+
=1 ;
+ ]
( + )+
3
=
=
1
2tan 3 2
3
3
(
)
+
+
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EXERCISE 9.2
INTEGRATION BY SUBSTITUTION
− − − − − −− −− − − 5
2 3 2
23.
2 +1
3
2
5
+1
5
3
2
+
3
3
3 3
( )
3 2
dx = 3
4
=
3
3 2
4
( )
+
( )
=
2 3
3
2
+
+
( )
2 +1
2 +1
For the 2nd term Let u = x2+1
− − =2
=
2
=
=
4
4
3 2 2
+
+
2
+
+
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EXERCISE 9.3 1.
INTEGRATION OF TRIGONOMETRIC FUNCTIONS
− − 5
5
7.
= 5
=5
= = =
5
=
3.
1
=
+
=
5.
2
+
=
+
1 sin 2
=
=
; Let u=
1 cot 2
1
+
2
2
1 2
=
2
=2 =2 =2
1
(
)
(
)
cos 3
)
sin 2
1
1+
cos 3
1+
+
;
=
2
+
)
=
+
2
)(1+
(cos 3 +cos 3
=
+
=
(1
=
+
5
1+
(cos 3 ) 1+
=
1
1+
.
1
=
5
1
cos 3
=
=
5
− − − − −
9.
=
+
=
+
=
+
=
2
2
+
+
2
1+
+ tan2 )
= (1+2
+ ( 1 + t a n2 )]
= [2 =2
= 2 =
+
|
sec 2
| +
|
| +
+
+
=2 =
+
+
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EXERCISE 9.3
11.
INTEGRATION OF TRIGONOMETRIC FUNCTIONS
6
15.
cos 2 3
Let u = 3x ; 2u = 6x
=3 ; 2
= = =
3
=
13.
+
= =
2
2
(2
)
= =
1
= =
+
+
2
2
2
− 2
.
2
=
2
=2
3
2
2
2
+
)
Let u = 2x
3
=
)(
2
=
1
2
(4
=
3
2
2
=
cos 2
2
4 sin 2
17.
=
2
1 2
+
3
3
Let u = 3x
− =3
=
1
=
3
=
3
=
3
+
+
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EXERCISE 9.4
INTEGRATION OF EXPONENTIAL FUNCTIONS
− − − − − − − − − 1.
2
2
=
dx
=2
;
(
)
=
2
1
+
2
1
=
2
+
(
=
)+
4
3.
=4
=
4
;
= 4;
(
4
4
=
)
1
=
4
=
;
1
=
2
=
2
=
=
2 ;
1
=
=
=
4
1
4
+
= cos
;
=
3
5.
3 2
=
3
=
= =
(
2
2
2
;
3
=
)
3
2 3
=
3 2
2 3
+
+
=
− − − − − − − − − − 7.
53
2
=3
= 5 ( = =
1
2
2
;
=
2 ;
2
=
)
5
2
1 53 2 2
+
5
+
=
+
9.
3 2
=(
=
=
)
6
+
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EXERCISE 9.5
1.
INTEGRATION OF HYPERBOLIC FUNCTIONS
−− − 3
1
Let u = 3
1
= 3 ;
(
= = =
=
3
)
=
3
+ c
3
+
2
2
1
2
Let u=1
= -2
=
=
7.
;
=
1
;
=
2
+
(
)+
− −
Let u =
1 2
1
1
2
2
;
=
1 2
; 2
=
= 2
= 2( =
+ ) +
=
2
= =
=
1
1
=
=
− − − − − − − − 2
2
3
=
3.
5.
1
(
2
)
2
2 1 2
(
+ )
+
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EXERCISE 9.6
APPLICATION OF INDEFINITE INTEGRATION
2
1. Given slope 3 2
=3
=
=
7. Given slope
2
+4
1
=
+
+1
−
+1 1
1
1
=
+1
2
2
2 =
2
2 =
+
2
2
+
5. Given slope
1
=
1
=
2
2
=
=
ln 2 2
+
+
+
2
+2 +2
+
2
1
+
=
=
4 1
ln1
− − − −− =
+
ln
+
3. Given slope
=
4
+4 +
3
, through 1,4
=
2
3
3
2
=
+4
3
− − − − − − − − −−− − 2
+4
2
= 3
+4
=
4
1
=
4
1
ln
4 ln
+
1 4
4+ +
=0 4
=0
=
9. Given slope
, through 1,1
=
1 2
=
1 2
1
=
+
2
2
1
=
2
When
+
=1,
;
2 1 =1+ 2
1
=
2
=
=1
+
=1
2
+
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EXERCISE 9.6
APPLICATION OF INDEFINITE INTEGRATION
11. Given slope
−
− − − − −− 1
=
2
, through 1,2
v = -32t + vo when t = 1 sec, s=h=48ft
2
=
1
2=
1
2=
48 = -16(1) 2 + vo(1) + c2
2
1
=
h=-16t 2+ vot + c1
64 - vo = c2
+
When t = 0, s = 0, c 2 = 0
+
s = -16t 2 + vot when t = 1 sec, s = 48
1+
s = -16t 2 + c1t
=3
1
=
=
48 = -16(1) 2 + c1(1)
+3 x
c1=64
1+3 +
s=-16t 2 + 64t
=
v = -32t + 64 @ max, v = 0
13.
0 = -32t + 64
a=-32 ft/sec2
32t=64 a=-2
− − − − =
32
=
32
t = 2 sec s = -16t 2 + 64t s = -16(2)2 + 64(2) s = 64ft
v=-32t+c =
32 +
=
1
( 32 +
1)
s=16t 2 + c1t + c2
when t = 0, v = v o v=-32t + c 1 vo= -32(0) + c1 vo =c1 DIFFERENTIAL & INTEGRAL CALCULUS |
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EXERCISE 9.6
APPLICATION OF INDEFINITE INTEGRATION
15.
a = 32ft/sec2 a = 32
= 32 =
32
v = 32t + c 1
= 32 + =
1
32 +
1
S = 16t 2 + c1 + c2
when t = 0, v = 0 c1 = 0 v = 32t when t = 0 , s = 0 c2 = 0 s = 16t 2
400
=
=
16
20 4
t = 5 sec v = vt
*since it is a free falling body, its velocity is ( - ) vt = -32t vt = -32(5) vt = -160 ft/sec
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EXERCISE 10.1
PRODUCT OF SINES AND COSINES
−− −− − − − − − − − −− − − − − − − −
1. ʃ sin5 sin
=
2sin
sin
= [cos
cos( + )]
=5
=
1 = ʃ [cos 5 2
cos (5 + )]
1 = ʃ [ cos 4 2 =
1 2
cos6 ]
[ ʃ cos4
ʃ cos6
1 1
1
2 4
6
= [ sin4
sin6 ] +
=
+
3. ʃ sin 9
3 cos
1 = ʃ [sin 9x 2
+5
3 + x + 5 +sin 9
1 = ʃ [sin 5 + 2 + sin(3 2 = 5 +2
5
=
;
1
1
1
2
5
3
= [ cos =
;
+
5
8)]
;
=5
3
=3
8
− − − −− − −− − − − − − − − − − − − − 5. ʃ cos 3
2
cos
+
+
+ cos (
1 = ʃ [cos 2
=3
=
)]
2
+
+
= 3
2
+
+
=4
= 3
2
=2
3
1 = ʃ [cos 4 2
cos4
+ cos(2
= cos 4
=
cos2
+
=
1 2
1
2
4
4
3 + sin 2
3
cos2
= ʃ ( cos 4 1
3 )]
4
3 = cos 2
= [
+
sin4
=
cos2 ) 1 2
sin2 ] + +
=3
3
=
]+
+
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EXERCISE 10.1
7.
PRODUCT OF SINES AND COSINES
− − − − − 4
8
3
= 2 ʃ [sin 8 + 3 = 2 ʃ [
+
8
11 + sin 5 ] = 11
;
=5
= 11 ;
11
= 2[ =
3
1
11
cos 11
=
;
1 5
=5
5
=
cos5 ] + +
9.
− −− − − − − − − − − − − 5
4 +
2
3
6
5 = ʃ [cos 2
cos ( + )]
=4 +
= 4 +
3
;
2
3
6
= 2 + /2 =2
+
6
= 4 +
+ 2
3
6
= 6 + /6
5 = ʃ [ 2
2 +
cos 2 +
5 = ʃ [ 2 5 1
= [
2 2
=
]
2
2
=
=
sin2 sin
2
cos 6 +
6
2
= cos 2 =
6 +
2
6
6
3
1
6
2
6
6
2
6
6
− − − − 3
2
2
6 +
2
3
12
6
1
12
1
6 ]
2
6 +
− −
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+
EXERCISE 10.2
POWER OF SINES AND COSINES
− − − − − −− − − − − − 3
1.
4
= =
4
;
4
2
(1
=
(1
=
(
) 2
2
2
4
4
+
6
2
)
8
+
)
4
2
2 6
6
5
7
8
+
)
9
5
+
9
=
+
4
3.
4
=
3
3
4
=
=
=
4
(
2
3
3
3
3
2
1
6
3
3
=
3
5
7
1
5
1
1 (
=
= =
= = =
=
7
+
2
2
2
) 2
2
2
1+
)2
2
1
=
+
21
(
=
3
3 )
3
4
5.
;
3
7
+
=
=- (
5
15
4
=
=
=
4
Let u = cosx
-
=
− − − − − − − − − − − 1
2
2
1
2 +
4
2
(
2
1+
2
4
1
1 ( 2
4
1
1
8
1
(1
8
1
8 1
2 +
2
1
2
4
2
2 + 2
2 +
(1
8
1
2
2
+
2 )
2
2
2
2
2
2
+
2
1+ 2
2
2
3
2 +
3
2 +
2 )
2 )
− − − − − 2
2
[
1
2
(
1
+
1
2
4 +
− −
8
1
2
2
8
3
+
1 2
2
1 6
2
3
2 ]
+
Let u = sin3x
=3
=
(
3
4
;
6
)
3
=
3
3
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EXERCISE 10.2
POWER OF SINES AND COSINES
− − − − − − − − − − − − − − − − − −
7. (
=
) 2dx
+
(
+2
=
2
+
1 2
+2
3
.
)
2
=
1 2
+2
2
2
2
+
=
=
2
+ (
1+
2
2
)
Let u = sinx
2
1
=
= =
2
2
;
=
2
2
2
1
=
2
2
3
1
2
+
3
=
=
1 2
+2
== -
2
2
1 2
3 2
+ + 2
2
3
+2 1
6
5
5
7
.
2
+
2
2
2
2
+
+ + 2
+
=
+
+
4
2 +
3
+
2
2 )2
3 +2
3
=
2
+
2
+ +
3 +
=
=2
3
+
9. (
=
2
+2
+
1
=
7
=
7
=
7
2
1
9
7
9
4 +
8
+
=
8
10
8
10
+
=
2
11.
= =
+
4
1+
=
u=sinx du=cosxdx
2
=
1
2
8
2
1
1+
2
+
8
+
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EXERCISE 10.3
POWER OF TANGENTS AND SECANTS
2
2
4 2
2
=
2
2
=
2
2 (1 +
= (
2
2
2
2
= =
1
2
2
2
+
=
)(
+
5
+
1 2
=
1 2
(1 +
=
1 2
(1+2
1 2
= (
;
4
2
2
5 2
;
+2
3
7
2 2
4 2
+
7
+
5 2
+
)2
2
+2
=
=
2
)
=
3
2
)
2
6
=
2
=2
+
1 2
2
2
5
3
= (
2
2 )
2 )
;
4
+
3
1
2
2
4
+
(
4
2
=
= (
2
2 +
=
2
2
2
2
9 2
+
2
=
9 2
+
4
+
11 2
11
+
2
)
)
;
2
=
→ − ____
1
.
=
2
3
2
+
+
− − − − − − − − 2
3
2
2
:
2
=
2
2
=
2
=
2
=
2
=
2
2
2
2
2
2
2
2
2
(
2
(
2
2
1)
2
2
4
=
1)
2
2
2 2
2
(
2
2
)
dx
,
=
(
= (
+
2
"
)2
+2
+2
+
2
=
+2
+ (
2
=
+2
2
+
=
=
+1
+
+
)
1)
+
+
2
)
+
+
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"
EXERCISE 10.3
POWER OF TANGENTS AND SECANTS
− − − − − −− −− − −− − − − − − 3
(
3
43
=
43
=
4
3
=
2
3
2
=
4
2
1 3
= =
3
2
=
=
4
(
1
3
3
3
2
2
3 )
=3
)
2
3
33
3
3
+
13
3
4
3 )
2
;
3
3
+
1
=
4
3
3 +
=
3
3
3 (1 + 4
= (
3
)4
+
2
2
3
3
3
− − − − − − − − − − − =
3
=
2
= (
2
1 2
3 2
1 2
= (
3 2
1)
3 2
=
;
)
=
=
= (
1 2
3
=
=
2 2 3
2
3 2
)
1 2
+
+
+
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18
EXERCISE 10.4
POWER OF COTANGENTS AND COSECANTS
− − − − − − − − − − − − − − − − − − − − − =
4
4
4
(1 +
4
= (
=
=-
6
;
4
(
5
5
2
+
=
2
)
=
1 2
3
2
=
1 2
3
1+
2
)
2
=
6
+
;
2
=
1 2
=
)
7
=
5
3
=
4
2
4
4
=
3
=
3
= [
3
3
4
2
4
2
2
4
=
=
2
4 (
= (
=
1)
3
4
4
5 2
2
3
3
2
3
3
3
2
3
3
;
4
+
2
4
2
+
2
=
+
1 4
+
1)
2
1 4
2
=
1
1 2
3 1 3
3 2
3 2
+
3
+
7 2 7 2
5 2
+
=
4
4
4
4
4 )
2
(
1
3
4
4
4
4
4
1
1
2
3
+ c
3
=
2
3
3 +
=
+ c
+
=
3
7
+
= -
=
4
3
+
4 ]
4
4
4
(
4
4 )
+
+
DIFFERENTIAL & INTEGRAL CALCULUS |
Feliciano & Uy
19
EXERCISE 10.4
5
= =
=
POWER OF COTANGENTS AND COSECANTS
5
2
=
8
2
5
2
2
5
3
2
4
2
2
2
2
\
2
2
2
2
1
2
6
2
− 3
=
2
2
2
− − − − 4
1
=
2
−
=
6
2
6
1+
6
2
2
2
4
+
2
− − − − − − − − − − − − − − − − − − − − − =
4
2
6
=
2
= =
=
2
2
6
2
2
2
2
:
2
7
7
=
2
=
2
=
4
+
6
1
4
+
5
2
2 5 5
( )
2
2
2 2
2 +
2
=
2
1
1
4
2
2
=
2
2
2 +1
2
=
=
2
2
=
1
2
3
5
5
=
5
+
3
3
+
3
+
3
+
+
3
+
=
+
+
+
DIFFERENTIAL & INTEGRAL CALCULUS |
Feliciano & Uy
20
EXERCISE 10.5
−
TRIGONOMETRIC SUBSTITUTIONS
2
2
4
− − − − − − − − = ;
= 2 ; 22
2
=
=
2
=2 =2
2
=
=
4
4
=2
= 2[
=
2
2
2
=4
=4
2
1
2
2
1
2
2
1
2
2
2
+ ;
=
( ) 2
]2 + C
+
− − ; 9 2 +4
=3
=2 =
3 =2 =
2
;
3
=
2
9
9
2
2
+4
+4
2
3
=
2
2
3
=
=
+4
2
2
=
2
9
=
=
=
2
3
2
=
3
2
=
=
=
2
1
1 2 1
1
2 1 2 1 2
[-
|
+
|] +
+
DIFFERENTIAL & INTEGRAL CALCULUS |
+
Feliciano & Uy
21
EXERCISE 10.5
TRIGONOMETRIC SUBSTITUTIONS
− 2
9
− − − − − − − − − − 2
=
2 3
9
2
=
2
9
=
;
9
2
=3
=
=3
=3
=
=
)2 3
(3
9
)2 3
(3
2
9
2
1
2
=
2
(1
)
2
= =
2
2
− 9 4 2
3 2 2
2
= 3;
=2
=
=
2
; 2 =3
3 2
=
3
3
=
− 2
1 2 ( 3
=
= = =
3
(
3 2
3 2
(3
3 2( 2
)
)2
2
9
2(
)
)2
(
3
)
9
4
2
)
2
= =
+
DIFFERENTIAL & INTEGRAL CALCULUS |
Feliciano & Uy
22
EXERCISE 10.5
TRIGONOMETRIC SUBSTITUTIONS
;
2 +4 2
=2
2
=
= = =
2
=
2
2
=
9
=3 ;
=
= =3
;
=
3
=3
;
=
3
−
x
2
9
+4
+4
2
2
2
2
1
=
4
16
2
8
2
8
1+
8
2
2
1
1
8
2
+
1
+ (2) 4
+
3
2
2
4
8 2
2
+4
2
2
1 1
− 11.
2
2
1
=2
x
=
=
=
;
+4
=
=
=
2
2
4
= ,
;
=2
2
:
2
2
=
1 8
2
− − − 2
=
=
= =
3
2
=
9
9
=
; 3
=
2
9
3
3
(3
)
3
1 3
+
+
+
DIFFERENTIAL & INTEGRAL CALCULUS |
Feliciano & Uy
23
EXERCISE 10.5
TRIGONOMETRIC SUBSTITUTIONS
− ∅ ∅ ∅ ∅ ∅ ∅ ∅∅∅ ∅ − ∅ − ∅∅ ∅∅∅ ∅∅∅ − ∅ ∅ − ∅∅ ∅− ∅ ∅ ∅ − ∅∅ ∅− ∅ ∅ ∅∅ −− − 2
.
16
(
=4
=
;
=4
=
;
= 64sec 4sec 3 ; 2
=
=4
=4
=4
=4
=4
= 4(
=
4
4
=
=
)
3
= ;
3
3 2
=4
16
4
3
( 4
;4
2
=
(4
16
))
(64sec 3 )
tan4
sec ^2
(sec 2
1)^2
sec 2
sec 4
2sec 2
+1
sec 2
sec
4
2sec
2
+1
sec 2
sec 2
2 + 1/ s ec ec 2
2 +
1 2
+
+
+
− − − − − − ∅ − ∅ − ∅ ∅ ∅∅ ∅ −∅ ∅ ∅ − ∅ − − ∅ − − ∅∅∅∅ ∅ ∅ ∅∅ ∅ ∅ ∅ − − .
2
5
3
2
12 + 4
=2;
5
12 + 4
=2
9
=2
=
3=2
3=2
2
4
3
; 2
2
2
; 2 = 2
+3
=2
=
=
2
2
2
=
2
=
= = =
3
2
2
=
=
3
3
2
4
2
=
2
(
3
2
4
)
2
2
1 4 1 4 1 4
;
=
2
3
2
+
DIFFERENTIAL & INTEGRAL CALCULUS |
Feliciano & Uy
24
EXERCISE 10.6
ADDITIONAL STANDARD STANDARD FORMULAS
− − − − .
2
Let:
=
+ 25
=5 =
=
+
.
4
1
Let:
=
2
− − − − − .
36 36
2
9
=6
Let:
=3 =
3
1 3
=
36 36
3 2 1 3
=
3
2
2
9
36 36
9
=
2
+
+
1 3
1 3
+
3
8
+
6
8
2
+
+
=1
2
=
=
1
2
2
1
=
+ +
.
49
Let:
25
2
=7
=5
2
=
.
16 16
Let:
=5
2
+ 25
=4 =
4
=
=
1 4
16 16
2
+ 25 +
1 52
4 + 16 16
2
+ 25 +
4
2
+
4
2
+
+
+
+
=
+
+
DIFFERENTIAL & INTEGRAL CALCULUS |
Feliciano & Uy
25
EXERCISE 10.7
INTEGRANDS INVOLVING QUADRATIC EQUATIONS
− 2
.
− − − − − − − − − − − − −− − −− 3 +2
2
3 =
2
3 = 3
3
2
9 4
1 4
1 4
1 4
2
1 2
=
=
2+
3
=
2
1
2
4
=
3 2 ) 2
(
=
=
9
=
2
=
2
2
2
1 2
− − − − − − − − − − − − −−
2
=
3 1 2 2 3 1 + 2 2
+
1
2
+
+
+
2
2
=
2 +1
1 2
+
2
=
=
2
=
4
2
+
1
+
2
=
1
1 2
,
1
=
2
1
1 2
2
1 2
+
=
+
2
.
3
=
4
+1
+ 1,
=2
=
2
2
2
2
= =
+1 2
=
3
2
2
2
=
2
+
2
+
4
+1
2
2
+
DIFFERENTIAL & INTEGRAL CALCULUS |
+
2
+
+
Feliciano & Uy
+
+
26
EXERCISE 10.7
INTEGRANDS INVOLVING QUADRATIC EQUATIONS
− − − − − − − − − − − − −− − −− .
2
8 +7
Completing the square 2
8 =
2
7
8 +16=
=
4
2
4
2
=9
9=0
4)2 + 9
(
=3 ;
=
= =
=
2
+
2
+
1
4 3
+
4+3
+
3+2
9.
2 +9
3
=
2 +9
=3
+
2 +9
=
2
2
2 +9
+2
3
4
1
2 4 2 1
=2
3 4 2 1
3
4 2 1
2
= 4
=2
8
=
|
4 2 1
1;
=
8
1
4 2 1
2
4 2 +1
3[
+ ]
|
+
+
3
13.
(2 +7) 2 +2
+5
2 +2 +5
=
2 +2
+5
2 +2
=
2 +2
+5
2
=
=
=
+
|
+5
( +1)2 +4
+2 +5;
1
+1
2
2
+
= (2
2)
+
+ |+
+
+
2 +9
+9;
1
=3
=
3
2
2
1
6
2
4
=
=
7+16
− − −− − − − − − − − − − − 11.
+2
+
=2
2
+
+
DIFFERENTIAL & INTEGRAL CALCULUS |
Feliciano & Uy
27
EXERCISE 10.7
INTEGRANDS INVOLVING QUADRATIC EQUATIONS
−− − −− −− − − − −− − − − − − − − −− − − − − (
15.
3)
4
2
=
1
2
=
2
4
=
2 ;
4
2(
=
2)
=
2
2
=
4
(2
)2
4 (2
+
=
2
2
2 4
; 4
2
2 4
=
4
2
4 +20
2
4 +20 2
)2
2
+
+
= 2[
2
4 +20
17
2
2
4 +20
4 + 20 ;
4 +20=
= 2[
=
2(2 +4+17)
=
2 +4
=
2
4
(4 +9)
=2
2
4
− − − − − − − − − − −− − 19.
2
2
17
= (2
2
4)
+ 16
] 2 2 +16
2
4 + 20 + +
17 1 2
+
( ) Arctan 4
2
4
Arctan
+ ]
+
− −− −− − − −− − −− − − − − − − − − +3
17.
2
8
4 +7
=
2
8
4
=
8
=
=
= =
= -
16
2
+7
2
8
2(
4)
2 8
2
(4
)2
+7
2
8
;
=
; 8
16 (4
+
8
2
2 8
2
2
)2
+
DIFFERENTIAL & INTEGRAL CALCULUS |
Feliciano & Uy
28
EXERCISE 10.8
ALGEBRAIC SUBSTITUTION
− − − − − − 2 3
=
=
3
=
2
=3
3
2
=3
=
1
=
=3 =3
|
=
|
10
3
5
3
9 6 10 4 30
+
+
(
)
+
|+
1 4 1 2
4
=3
11
9
8
)
9
= 3[
= 3[
3)
8
(
8
10
9
10
9
10
9
10
3
4
=
4
3
+ ]
2
1
1
=
11
5
=3
=
=
12
( 4
1
+2 2
=
− − − − − = 12
3
+2 4
3
− 1
− −− − − − − − − − − − − − 4
1|+
( 3
=
7 4
= +2 4 = +2 4 =2 = 4 3
+
=3
3
5 6
1
=
=
− − −
3
]
+1
+1
=
4
=
4[
=
4[ +
=
[
+
]
+
+
+ ]
+
DIFFERENTIAL & INTEGRAL CALCULUS |
Feliciano & Uy
29
EXERCISE 10.8
ALGEBRAIC SUBSTITUTION
− − − − − − − − − − − − − 4+
;
=(4+
2
)1/2
4
4=
12
=
=
=
4 5
3
16
5
(4 +
4+
= =
15
3
4
(4 +
)3/2 +C 16
4+
3 2
12 4 +
48+12
80
80
15
4+ +
3 2
+
3
15
= 4+ 4
16
5
3 2
+
3
)5/2
3 2
= 4+
=
16
2
16
5
=4
3
= 4
16
4
4
+16=
4)2 =
3
4
=
2
(
2
8
12+3
+
− − − − − − − 1 3
+4
9.
=
+4
3
=
=
=
3
4
3
4
7 3
3
7
3
=
+4 7
+4
3
=3
6
=3
3
;
4;
3
=
1 3
4 3
2
3
+
7 3
3
+4
4 3
+
4
=
3
+4 3
+4
7
+4
1 3
=
7 + +
+
+
20 +
+
DIFFERENTIAL & INTEGRAL CALCULUS |
Feliciano & Uy
30
EXERCISE 10.8
ALGEBRAIC SUBSTITUTION
4
2 +1
1 2
2
= 2 +1; 2
2 =
1
=
=
=
=
2
2
2
;
=
1
1+
4
4
1
2
2
2
2
=
2
2
2
2
2
2 +
1
2
2
2
1
2
2 +1+ 2
=
+
− − − +
+
+
+
5
+
=
=
=
= =
=
2
2 3
3
4+
3
2
5
2
4
( )
2
4
2
2 3
2
3
8
=
1
4
2
=
2
2
2
2
3
2
3 4
2
2
=
2
=
2
2
;
2
3
=
2
3
=4+
2 3
4
3 4
2
2
1
=
2
4
=
=
1
=
3 2
4+
=
4
=
=
1 ;
=
=2 +1 2
x 5 4 + x 3 dx
.
− − − − −−− −− −− − − − − − − − − − − − − − − 11.
− − − − − − − − − − − − − − − − − − − − 2
+2
4
3
1
2
3
2
3
8
15
6
3
8
5
5
2
8
5
1 2 3
4
3
+
9
4+
+
3
3
40 4
+
45
3
4+
6 4+
3
40
45
3
4+
24+6
3
40
45
16+6 3
4+ 3
45
+
+
+
+
+
DIFFERENTIAL & INTEGRAL CALCULUS |
Feliciano & Uy
31
EXERCISE 10.8
ALGEBRAIC SUBSTITUTION
− − − − − − − − − − − − 3
3
3+1 2
(4 +
=2
2
Z= 4+ 2
2
= 4+
2
X= 4 1
dx =
2
= (4 =
=
=
4
4
5
28
5
)(
)
6
+
7
+
+5
+
7
7
+
35
28( 4+ 2 )5 +5( 4+ 2 )7 35
2
4+
5
4+ 2
5
( 28+5(4+
( 28+20+5 2 ) 35
(
ʃ − ʃ ʃ − − −ʃ − ʃ −ʃ ʃ − − − 1
4
2 +1
=
;
2
=
2
=
2
+1= =
+1=
&
1
3
2
= =
. =
=
2
(
1)
=
1
2
= 1
3
=
3
3
=
=
3
+
+
+
+C
35
+
=
3
)(
5
4
=
=
(-2zdz)
1 2 )2
(4
=
1 2
2
4
2
=-
2 2 )
2
)
+
+
)
+
DIFFERENTIAL & INTEGRAL CALCULUS |
Feliciano & Uy
32
EXERCISE 10.8
ALGEBRAIC SUBSTITUTION
− − − − − .
(
=
1
2 (81
;
) 4
+
1
=
2
2
1
2
1
81 +
4
2
81
4 +1 6
3
=
81
4
+1 4
= 81
=
=
=
3 4
+1;
= 324
1
324 1
81
81
3 4
4
+1 +
1 4
+
+
3
− − − − − − − − − − − − − − − − − 3 )1/3
(
4
1
=
, 2
1
=
1
1
3
=
2
1
4
1
2
1 3
3
=
2
1/
4
1
( 2 1) 3
=
(
2
)
4
2
=
1
1 3
=
=
=
=
=
1
2
3
(
4 3
4 3
8
)+c
2
8 3
1
1 3
2 1
2
1
2
1
1
4 3
4 3
+
+
=
DIFFERENTIAL & INTEGRAL CALCULUS |
Feliciano & Uy
33
EXERCISE 10.9
INTEGRATION OF RATIONAL FUNCTIONS OF SINES AND COSINES
− − − ∶
− −
.
1+
=
2
1+ 2
1 2 1+ 2
1+
=
= = =
1+
2
2
2
4+2 +2
=2
+
+
=
2
2
=
1 2
+
7
+
=
2
2 +1
7
7
7
+
2
=
2
+
1
2
=
2
2
1+
1+ 2 2 4+2 1+ 2
4
,
=
2
+
7
7
+
1 2
1 2
+
2
+
2
=
4
4+4 2 + 4 1+ 2
+4 +4
=2
+
2
3 2
1 2 2
+
1+ 2
2
2
+
=
;
:
=
+
1 2
;
=
3
.
=
2
2
=
=
2 2 +3+3 2 1+ 2
1+2
2
4 4+ 1+ 2
=
1+ 2
− − − − − − − 4+2
=
2
2
=
=
2
2
=
1+ 2
2 1 2 + +3 1+ 2 1+ 2
=
2
1+2 2 2 +3 1+ 2
1+ 2
=
+3
2
=
2
1+ 2 + 1 1+ 2
+
2
=
+
1 2 3 2
+
3
= 2
2
+
2
=
2
+
=
4
+
=
1
2 +1
3
+
3
+
=
2
1
2
2
+
1+
2
1
2
=2 +
+
.
2
2
= 1,
2
1+
=
+
2
1+
1
=
+
+
+
+
DIFFERENTIAL & INTEGRAL CALCULUS |
Feliciano & Uy
34
=
EXERCISE 10.10
INTEGRATION BY PARTS
− − −− −− − −− − − − − − − − − − − − − −− − − − − − − ;
=
=
=
=
=
= =
.
2
+
;
= -
2
= -
2
=
2
;
=
;
=
2
;
= -
;
2
2
= -
2 +2
4
2
=2
2
2
;
=
;
= -
2
2[-
-
2
2
2
4
2
2
2
2
2
+
5
=
+
2
;
=
=
=
2
=
2
=
2
=
=
2
2
= -
=
+
;
=
;
=
2
2
1+ 2
2 1+4 2
2
1+4 2
1 4
+
+
DIFFERENTIAL & INTEGRAL CALCULUS |
Feliciano & Uy
35
EXERCISE 10.10
INTEGRATION BY PARTS
− − − − − − − − − − − − − − − − 3
;
2
=
;
=
=
=
2
=
2
= =
=
3
2
+
=
2
;
1
=
2
2
+
=
1
+
2
1
1
4
8
2
+
1 8
4
8
+
2
(
1
2
4
=
4
=
1
+
2
+
;
=
= - 1
2
= - 1
2
+
;
1 8
1
4 )
4 +
32
+
+
1
= - 1
=
+
= 2 1
3
;
+
+
=
=
3
+
3
2
1
2
2
(- 1
;
=
;
=
2 )(
1
1
2
2
)
+
+
DIFFERENTIAL & INTEGRAL CALCULUS |
Feliciano & Uy
36
EXERCISE 10.10
INTEGRATION BY PARTS
− − − − − − − − − − − − − 1+
;
=
1+
1
=
1+
;
=
;
=
2
= -
1+
+
= -
|1 +
|+
= -
1+
+
1
= -
1+
+
+
=
(
+
;
+1)2
=-
=
+1
+
=
3
3
3
3
3
3
=
1
= -
+1
+
=
1
;
2
2
=
3
;
=
3
3
3
9
3
1
1
2
9
2+ 2
2
9
2
;
)+
+
+
=1 ;
=
;
=
;
=
;
1
2
=
3
(
3 1
;
( +1)2 1
2
3
1 3
=
1
1
+
;
=
1
+
3
1
;
3
+(
3
+
+
;
=
1
1
+
=
1
1
+
+1
;
1
1+
=
2
=
2
1
+
=
=
1+
)
2
3
= (
+
)+
DIFFERENTIAL & INTEGRAL CALCULUS |
Feliciano & Uy
37
EXERCISE 10.11
INTEGRATION OF RATIONAL FUNCTIONS
− − − − − − − − − − − − − − − − − − − − − − − − − −− − − − − −− 12 +18
+2
+4 (
1)
12 + 18
+2
+4 (
1)
12 + 1 8 =
+4
12 + 1 8 = (
2
2
12 + 1 8 =
=
( + 2)
+
( + 4)
1 +
+3
4)+
+3
4 +
+
+2
2
+
2
+
(
1)
1 +
( +2)
3
( +4)
1=
(
(
+
1)
1=
+6
+8
2
2 + =0 +6 = 12 + 8 = 18 =1 = 3 =2
1)
+
4)
+
(
4)
4 + (
1=
+ 6 + 8)
2
+
+
1 (
2
2 +
+ 3 + 4 +
+
2
2 + (
2
=
+ 2 ( + 4)
1)
4 +
+
=0 =1
4
=
=
1
=
3
(
=
1 3
1)
1 3
1
+
3
(
1 +
4)
1 3
4 +
=
DIFFERENTIAL & INTEGRAL CALCULUS |
Feliciano & Uy
38
EXERCISE 10.11
INTEGRATION OF RATIONAL FUNCTIONS
−− −− − − − − − − − − − − − − −− − − − 6 2 +23
9 ( 3 +2 2 3 )
2
6
+ 23
9
+3 (
6
2
6 6
2)
+ 23
9=
2
+ 23
9=
2
+ 23
+
( + 3)
+
+3
2
9=
(
1)
1 +
+2
3 +
1 +
2
+ (
( + 3)
2
+3 )
+
+ =6 2 + 3 = 23 3 +0 +0 = 9 =3 = 2 =5
=3
2
( +3)
+5
+
3 +5 2 +9 2 +5
3
2
+5
(
1)
+
+
+7
+4
+9 +7
+ 4 ( + 1)
By division of polynomials, 5 +7
+ 4 ( + 1)
5 +7 =
=
( + 4)
+
( + 1)
+ 1 + ( + 4)
= 4, 13 = 3 = 1 2 = 3
13
= =
+
+
2
3
( + 4) +
+
+
3
( + 1) +
+
DIFFERENTIAL & INTEGRAL CALCULUS |
Feliciano & Uy
39
EXERCISE 10.11
INTEGRATION OF RATIONAL FUNCTIONS
− − − − − − − − − − − − − − − − − − − − − − − − −− − 2 +1
3)2
2 (
2 +1 =
(
2)
2 +1 =
+
3
2
2 +1 =
2
(
3)
+
+
3)2
(
3
2 +
2
6 +9 +
2
5 +6 +
2
+
6
=0
5 +
9 +6
=2
2 =1
=5
=
5
=7
=
=5 =
5
2
2
5
+
3
5
+
(
7
+
3 +
3
2
7
3
)
DIFFERENTIAL & INTEGRAL CALCULUS |
Feliciano & Uy
40
EXERCISE 10.11
INTEGRATION OF RATIONAL FUNCTIONS
−− −− − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − −− − −− − − − − − − − − − − − − − − 2
5
(
2
5
(
1)
1)
=
+
(
1
+
1)
3
2
5=
2
5=
3
3
2
+3
2
5=
3
=3
2
2
2
5=
3
+
2
5=
+
1)2
(
+
1
2
3
2
+
1 +
3
+
2
2
2
2
2
+C
+
+
+
2
+
2 +
2
2
+
2
3
13
(
+
3
+
2
+
+3
+
+
+ 3 +
+
+
3
=0
2 +
3 +
=0
+
=
=2
5
=5
=
5
=5
=
=
5
5
+
(
=5
5
=5
5
=
1)
(
5
+
1)
+5
(
(
)
+
1)
(
+
3
1)2
2(
)
1)3
(
3
1)2
(
5
1
3
+ 1)2
(
3
(
1)3
+
+
DIFFERENTIAL & INTEGRAL CALCULUS |
Feliciano & Uy
41
EXERCISE 10.11
INTEGRATION OF RATIONAL FUNCTIONS
3 2 +17 +32 3 +8 2 +16
3
2
+ 17 + 32 ( + 4)2
3
2
+ 17 + 32 ( + 4)2
=
+
( + 4)
+
− − − − =
2
+
=
( + 4)
+
+
( + 4)2
+ =3 8 + 4 + = 17 16 = 32 =2 =1 =3
3
+
( + 4)2
+
+
2 +1
1 ( 2 +2 +2)
3
− − − − − − − − −− − − − − 2 +1
3
2
1 (
+ 2 + 2)
=
(3
1)
2 +1 =
2
+2 +2 +
2 +1 =
2
+ 2 + 2 + (6
= =
=
5 2 5 2
+
(3
1)
3
1 +
+
+
5 2
5 2
+
2 +2 + 2
+2 +2
2 +2 3 2
1 +
+4 +2)+
3
3
1
1
+ =0 2 +4 +3 =2 2 +2 =1 5 = 2 5 = 2 = 1
(2 + 2) 2
2
+2 +2
+2 +2
2
+2 +2
+
2
+2 +2
+
DIFFERENTIAL & INTEGRAL CALCULUS |
Feliciano & Uy
42
EXERCISE 10.11
INTEGRATION OF RATIONAL FUNCTIONS
− − − − − − − − − − − − − − − − − − − 5 2
+17
+2 ( 2 +9)
2
5
+ 17 2
+2 (
=
+ 9)
+2
2
+
2
+
2
+9
5
2
+17=
5
2
+17=
2
+9 +2
5
2
+17=
2
+2
5
2
+17=
+2
+9 + 2
2
2
+
2
+4
( + 2)
+4 +
+ 4 +
+
+2
+9 +2
+9 +2 2
=
+2
=5
=4 +
=
1
= 9 + 2 = 17
+2
=5
4 +
2 +
2
=
=
1
1
9 + 2 = 17
=
=
2
4
=
4 +
=
1
2 +
=
11
2
=4
=
10
2 = 22
9 + 2 = 17 13
= 39
A=3
9(3)+2C=17
4B-5=-1
27+2C=17
4B=-1+5
2C=17-27
4B=4
2C=-10
B=1
C=-5
=
− − −
= 3
3
+2
+2
+
+2 +
1 2 2
5
+9
2 +9
+
5
2 +9
+
+
DIFFERENTIAL & INTEGRAL CALCULUS |
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43
EXERCISE 10.11
19.
INTEGRATION OF RATIONAL FUNCTIONS
4 2 +21 +54
− − − − ʃ − ʃ − ʃ − ʃ − − ʃ − − − − − 2 +6
+13
4
3
2
2
+ 6 + 13
2 +6 +
2
+ 6 + 13
2 +6 + 2 +
=
=
= 4
[
=
11
=
11
=
11(
=4 =
3
2 +6
2
2
2
+ 6 + 13
+ 6 + 9 + 13
+3
2
+ 13
1
+3
2
2
3 2
+ ( 11
|
|
2
9
2
=3
11
3 2
+ 6 + 13)]
9
2
)
+ 6 + 13|
+
2
=3
+
|+
11
+3
2
2
+
+
DIFFERENTIAL & INTEGRAL CALCULUS |
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44
EXERCISE 10.11
3 +7 2 +25
INTEGRATION OF RATIONAL FUNCTIONS
+35
2 +5
+2+
+6
9 + 23
2
+5 +6
9 + 23
+ 3 ( + 2)
=
+3
+
+2
9 +23=
+ 2 + ( + 3)
x=-3
9(-3)+23= A(-3+2)+B(-3+3) -27+23=A(-1)+B(0) -4=-A A=4 If x=-2 9(-2)+23= A(-2+2)+B(-2+3) -18+23=A(0)+B 5=B B=5
= =
=
− − − +2+ +2
+
2
+3
+
4
+
5
+2
+3
+
+5
+2
+
+
DIFFERENTIAL & INTEGRAL CALCULUS |
Feliciano & Uy
45
EXERCISE 10.11
INTEGRATION OF RATIONAL FUNCTIONS
− −− 2
(2
8
2 +2
3)(
+2)
− − − − − − − 2
A(
2
A(
3
+ 2 +2)+
2
2
+
2
2 +
+2 +2
2 +2 2
+ 2 +2)+
4
2
2
3 + (2
6 + (2
3)
3)
A+4B=1
2A-2B+2C=-1 2A-6B-3C=-8 1
A=-
2
1
A=
2
C=1
− − − − │ │ − − − │ │ │ − │ 1
(2
(2
=
=
1 2
3)
3) + 2
+
+
1
2 +2
2
2
1
2
2
+2 +2
2
+2 +2 + 2
3 + +
+
+
+2 +2
+1
+2 +2 +
+
2
+ 12
+1 +
+
DIFFERENTIAL & INTEGRAL CALCULUS |
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46
EXERCISE 10.11
INTEGRATION OF RATIONAL FUNCTIONS
ʃ − ʃ − − 5 +2 3
3
2 +1 3
5
=
6
+2
+3
2
=
4
+
2
3
3
+2
2
+
+1
+1
2
+
2
+1
2
=
2
2
+1
2
+
=
2
4
+2
2
+1 +
=
2
5
5
+4
3
+2
2
+
+1
4
4
+
2
4
:
;
=0
3
:4 + 2 = 2
;
=0
2
:2 +
;
=0
3 ;
=0
;
=0
:
=
+
+
+
=0
+
+
+1
+
2
3
2
2
2
2
+1
2
+1 +
+1 +
+1 +
3
2
2
3
3
+1 +
+2
+2
2
+
+
2
2
+
+1 +
+1 +
2
2
+
+
1
=
:2 + 2 + 2 =
2
+2
;
=0
+
+2
:2 = 1 =0
2
2
+
DIFFERENTIAL & INTEGRAL CALCULUS |
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47
EXERCISE 10.11
INTEGRATION OF RATIONAL FUNCTIONS
− − 4
+2
3
+ 11 2
(
[ +
2
(
2
2
+ 8 + 16
+ 4)2
+
+ 4)
+
(
2
+
2
+
4)2
(
2
+4)+
4
+8
A
2
+4
2
+
A(
4
+8
2
+16)+
2
=1
A=1
2
4
:
+2
3
:
=2
2
: 8A+8B+2D=11
2
][(
2
+ 4)2 ] 2
+
+4 ( )+ 3
+4
+
(2 )( ) + ( ) 2
2
+
B=0 C=2
X: 4C + E=8
D = 3/2
C : 16A = 16
E=0
2
=
+
=
+ 2
=
+
2
+4
+
3
2
2
(
2
+ 4)2
1 2
3
2
2
+
2 +4
+
+
DIFFERENTIAL & INTEGRAL CALCULUS |
Feliciano & Uy
48
EXERCISE 11.1
SUMMATION NOTATION
∗
− − − − =10
= 10
.
12
.
=1
=1
3
=10
=1
10 1 0 + 1
= 12
3
=
=10
2
1)( + 1)
=1
3
= 12
(
3
=
2
=1
4
=10
=10
3
=
= 3(100 121 )
+
=1
=
=
=1
10 2 10+1 2
10 10+1
− − ⋯ − − 4
2
=
=10
.
(12
2
+4 )
=
.
=1
=10
+4
=1
= 12
=10
=1
10(10 + 1)(2 10 + 1) 6
= 2 110 21 + 2 110 =
=
=10
2
= 12
+
+4
10(10 + 1) 2
=
9
2
+6 +1
=1
=9 =9
2
+6
10(10+1)(2 10 +1) 6
+
+6
1
10(10+1) 2
+ 10
=
.
+
+
+(
)
=
=
DIFFERENTIAL & INTEGRAL CALCULUS |
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49
EXERCISE 11.1
SUMMATION NOTATION
∆ ∆ ⋯ ∆ ∆ .
1
=
1
+
2
+
2
+
( )
=
. 14 + 24 + 34 +
⋯ +
4
=
=
⋯ .
1 1+ 2 2+ 3 3
+
+
=
=
⋯ .
3 1
+
3 2
+
3 3+
+
3
=
=
DIFFERENTIAL & INTEGRAL CALCULUS |
Feliciano & Uy
50
EXERCISE 11.2
THE DEFINITE INTEGRAL
− ∆ ∆ ∞ →∞ →∞ →∞ →∞ →∞ .
=0;
=
0
= =
=
2
3
3
24
8
6 3
1 0 0 0 2 3 + 2 +2 2 +
=
24
6 3
2
1
2[ ( ) 2
]
1
3
}
2 3
+1 2 +
2
1
2
1
2
6
3
2
2
+
3
0
2
+2
2
+
6
1
1
∆ − ∞ ∙ ∞ ∞ 5 2 1
( + 1)(2 + 1) 1
2 +1
=
=
3
2 +1
;
2
=
2
24
0
2
=
( )
6
=1
=
2
2
;
1
2 ( )
2
1)
2
=1
4
(
1
={
2
1
=
=
2
3
=
=
+
=0+
=
=0
2
=
=
1 2 0
=2
2
=
=
− − ∆ →∞ − →∞ − →∞ − − →∞ − − −
2 3 2 1
3
=
+3
5
1
;
=1+
4
4
=
4
=
=
=
=
=
=
4
(1 + ) 4
4
16
+
+
2
16 2
+
+3
+
3
( +1) 2
+3
=
DIFFERENTIAL & INTEGRAL CALCULUS |
Feliciano & Uy
51
EXERCISE 11.2
THE DEFINITE INTEGRAL
∆ →∞ →∞ →∞ →∞ →∞ 2
3
.
0
=
2
;
=
2
3
8
=
=
3
2
=
=
2
=
2
3
2
16
+1
4
4
4
2
3
4
(
2
(
4 4 4
2
+
+ 2 + 1)
8
3 +
4 2 3
=4+0+0
DIFFERENTIAL & INTEGRAL CALCULUS |
Feliciano & Uy
52
EXERCISE 11.3
SOME PROPERTIES OF THE DEFINITE INTEGRAL
− − − − 2
.
2
3
2 +1
.
1
3
3
=
+
3
=8
4+2
1+1
− − .
2
3
4
+
2
1
3
3
=
+
3
4
4
= 27
1+4
3
+1
+1
=2
1
= =
3
2
=
+
2
2
2
2
2
− − − − − − − − − − − − − − 3
3
1 2
2
1
10
2
5
= .
0 1
( 2 +2
1)
0
=
( +2 +1
1
1
1)
0
=
[
1
2
+1
+ 2]
0
=
7
.
3
1+
2
0
2
=1+ = 2
4
=
1
3 1+ 2 3
2
4
+1
1
2
+2
0
=
1
2
+1
= 2 ;
=
+1 2
2
= ( + 1)
+
=
=
DIFFERENTIAL & INTEGRAL CALCULUS |
Feliciano & Uy
53
EXERCISE 11.3
SOME PROPERTIES OF THE DEFINITE INTEGRAL
− ∙ − − − − − ∙ ∙ − − − − − − .
2
0
=
=
=
= =
0
1
+ ;
=2
;
2
=
2
0
1 2 1
0
2 1
2
ln
2
+
cos
0
2
ln
2
2
1
+
0+
+
=
2 1
+1 =
2
1
;
=
2 0 2 +4
2
4
=
+1
1 2
= 2
=
; =
=4 4;
= 0,
2
4
=0
2
1 2
4
=8
2
2
0
4
=2
2
1
4
0
2
2
0
u= x; du=dx; a=2
1 2
; 2
; 2
2 2
=2 = 1,
=8
=
2
+1
+
.
2
=
=
1
=
=
2
2
2
=
=
.
+
2
=
1
1
1+
2
2
2
2
=
1
1
=
.
0
= =
=
= 1
; ;
= =
1 0
1+0
=
1
DIFFERENTIAL & INTEGRAL CALCULUS |
Feliciano & Uy
54
EXERCISE 11.3
SOME PROPERTIES OF THE DEFINITE INTEGRAL
2
.
2
0
=
2
=
;
=
2
3
=
3
3
=
3
=
− −− −− − −− − .
=
2
6
4
6 1 6 3 6 5 (4 1)(4 3)( ) 2
(6+4)(6+4 2)(6+4 4)(6+4 6)(6+4 8)
=
6
.
2
− −− −−− − − −− ∅− ∅ ∅ − ∅ ∅∅ ∅ ∅∅∅ ∅ ∅∅ − − − 2
0
=
;
2
2
=
2
6
=2
2
6 1 6 3 6 5 2 1
=2
6+2 6+2 2 6+2 4 6+2 6
2
=
4
.
2
2
4
2
8
2
= =
1 2
1
2+2 2+2
2
1
4 2
2
=
2
.
3
2 2
4
;
=2
0
=2
−− −− −− .
=
=
2
7
(4 1)(7 3)(7 5)
7(7 2)(7 4)(7 6)
2
=
3
4
2
(4
2
2 2
(2
)
0
2
=
3
)2 2
0
2
8
3
4 1
4 3
=
2
0
=(
4 4 2
2
=
DIFFERENTIAL & INTEGRAL CALCULUS |
Feliciano & Uy
55
EXERCISE 12.1
AREA UNDER A CURVE
2
=3
;
=1
=2
− − =
=
−
1 ;
2
= = =
2 3 2 1 2 3 1
= 2 =
3
1
.
3
=2
1
2
1
=1
=
1
− − − − − − =
2 1
1
2
=[
]
1
= {[ =
2]
[
1]}
2;
,
,
=
.
DIFFERENTIAL & INTEGRAL CALCULUS |
Feliciano & Uy
56
EXERCISE 12.1
AREA UNDER A CURVE
– − − − − − − − − − − = 3
,
=2
=4
+
=3 &
=
0
4
= 3
2
= 3[
]
= 3[4
4
4]
= 3[4
4
4
3[2
2
= 3[8 2
2 2
= 3[6 2
2]
= 6[3 2
1]
=
[
]
2
2]
2 + 2]
2]
=9
=
3 3
4
2
=
;
=
3
3 0
3
3 3
= 3
.
− − − − 7.
− − − 2
3 2
= 3 3 =
=3
0
2
.
2
=
DIFFERENTIAL & INTEGRAL CALCULUS |
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57
EXERCISE 12.1
AREA UNDER A CURVE
2
=4 ,
=1
− −
=4
.
= 1,
= ,
= 2,
=0
4
=
4
1
4
=
1 2
4
1
=
= =
=
8 3
3 4
8(4)3/2
8(1)3/2
3
3
64
8
3
3
.
= 1;
=
( )=1 =1 ;
=1
; (1,1)
− 2
1=
1
1
= ( =
)
2
1= 2=
=
1 2
2=
1
2
.
1
2
1 1
1 2
.
= 1 +
=(
2
+ )
.
DIFFERENTIAL & INTEGRAL CALCULUS |
Feliciano & Uy
58
EXERCISE 12.2
AREA BETWEEN TWO CURVES
− − − − − − − − − − − − − − − −− −− − − − − − − − − − −− − − − = 2; =2 +3 =2 +3 2 =2 +3 2 2 3=0 3 +1 =0 = 3, = 1
1.
=
3 1
= [
2
2
2 +3
3
+3
3
]3
-1
(3) 3
= 32 + 3(3) = 9+
3
( 1) 3 3
5 3
=
.
2
3.
( 1)2 + 3( 1)
=
1
;
=
3
Y 1=Y 2 3 2= 1 2 6 +9 = 1 5 2 =0 =5 , =2 =5
3=2
2
=
+3
(
2
+ 1)
1
=
2 1
+2
2
=
=
=
2
22 2
10 3
2
2
3
+2
3
+ 2(2) +
7 6
-1
23
1 2
3
2
=A=
27 6
=
+ 2( 1)
( 1)3 3
5. y = x 2 ; y = 2
−
x2
− − − − − − − − − − − − − − −− − − − = 2 ; (0,0)
=0,
=0
2
2
=2(
)
:
y1= y2 2
2
=2
2
2
2+
=0
(2 + 2)(
1)
2 +2=0 1=0 2 2 = =1 2 =
1
=1
=[ 1
2]
1
=
2
(2
2
)
1
1
=
(2
2
2
)
1
2 3
= 2
=2 =
3
2 3
+2
2
= 2 2 3
3
=
2
[ 2+ ] 3
12 4 3
.
.
DIFFERENTIAL & INTEGRAL CALCULUS |
Feliciano & Uy
59
EXERCISE 12.2
AREA BETWEEN TWO CURVES
− − − − − − − − − − − − − − =
7.
x 0 90 180 270 360
;
y 0 1 0 -1 0
=
x 0 90 180 270 360
;
=
=
4
3
=
11.
2
= 8,
2
=3
y 1 0 -1 0 1
,
=0,
=0
, 0=3
2
=0
2
2
=6 (
)
:
2
2
=
= [-
]
y1= y2
4
= [-
4
]
[-
2
]=
2
3
2
=8
3
1
2
=
=
=
4
4
3
=8
=
2
= 1
2
8=0
2
3
8
=2
=
.
=2
= 8 , (2,8) =
2
9.
= 4 ,
=
8
=
2 +4
2
2
3
,
=
8
2
=
2
=
(-2,-8)
4
2
+ 4 = 32 2 8 2
= 4.95
2
+4
2
4
=[ 1
2]
2
=
(8
3
)
0
=
=
DIFFERENTIAL & INTEGRAL CALCULUS |
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60
EXERCISE 12.2
AREA BETWEEN TWO CURVES
− − − − − − − − − =2 +1 ,
13.
=7
,
8
=
2 +1
7
2
8
=
2 +1
7+
2
= =
= =
8 2
3
3
2
2
3(8)2 2
6
8
6
2
6(8)
3(2)2 2
6(2)
=8
− − − − − – − − − − 3
=
=
(
,
=
;
=
1)
2
1
=
3
[(
)
(
)]
1
=
3
1
3
=
=
=
1
;
3 2
;
=
3
3
(
3
1
=
3
=
3
=
;
=
3
3
;
=
=
;
=
2
3
)
[
1
1
=
[
(
)]
1
]1
.
DIFFERENTIAL & INTEGRAL CALCULUS |
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61
EXERCISE 12.2
AREA BETWEEN TWO CURVES
− − − − − 2
17. 2
=2
2
=2
2
2+ 2
;x=
2
=4
=4
2
=
2
,
4
2
=
2
=
0 = 0 ; (0,0) 2
2
1
=
:
X 1 = X 2 2
2
=
2
4
2
4 4 2 2
2 2
2
+
2
=2 2 2 2 2 2 =2 2 3 = 2 = 2
+2 2 2 3 =0
3
=0
− − − − − − − − − − − − − − − =
4
= ( 4 )2
=
16
2
2
=8
2
=
=
(
[
12
=
12 3
= = =
4
12
2
4
-a
3
+
3
12
2
3
12
3
+
− − 2 3
12
+
2 3 4
=
)3
(
12
3
2
]
3 a
2
2
+
2
)
2
3
2
2
4
+
2
+
4
2+ 2
3
=
3
2
= =±
3
2
2
+
+
(
)
4
2(
12
3
4
2 3 +6 3 12
4 3
12
A=
a2 3
sq. units
2
X 1 = X 2=
2
=
2
=4
=
=
(4 )2
16
2
2
2
=8
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62
)3
EXERCISE 12.2
AREA BETWEEN TWO CURVES
− − − − − − − − − − −− − −− − −− − − − − 2
.
2
1=
=
+1
;
1;
=2
2
=1
=
2
)
2
1= 2 ;
1=
0=
+
2=0 ( 1)( + 2) 1=0 +2=0 y=1 y=-2 =0 =3 = 2
= 2, = 1, = 2, = 3,
= 5 =0 =3 =8
2
2
2
1
2
2
= 2
2 1
+1
2
3
(1)2
(1)3
1
2
3
=2 =
1
1
2
3
+4 2
4 + 4(4)
2
2
4
16=0
8
+2
8=0 +2=0 = 4; = 4(1, 2)
=(
=
2
+4+2
2
(4, 4)
2
− − −
2
2
2
2
= 2( 2)
+4
=
2
2
− − − − − − − − = 2(1)
+4
2
=
4
3 1
2
= 2
1
4
2
2
= 1
1
4
4
= (concave to the right)
2
1
=
2 =
4
1
=4
0,0
1
1
;
0=0
;
=
2
=
1
= 1,
=4
2
=0
=2 (
2
4 =
2
2
− − − − − − − − − .
=1
;
= 0;
=1
( 2)2
( 2)3
2
3
=
1)
2
4 2
+4 2
2
4
.
8 3
.
DIFFERENTIAL & INTEGRAL CALCULUS |
Feliciano & Uy
63
EXERCISE 12.2
AREA BETWEEN TWO CURVES
− 2
23.
=
+4,
2 +1=0
− − − − − − − − 3
=
2
2
1
4
1
3
=
2
3+2
1
3
2
= 3 +
3
3
1
=
2
=
25.
,
= ,
=2
− − − − − 2
2
=
0
2
2
=
2
4
=
2
0
2
1 2
+1
=
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64
EXERCISE 12.4
VOLUME OF A SOLID OF REVOLUTION
− − − −− − − − 2
=
1.
2 ,
=2
,
2,
0=2
= 12
2 ;
=1
;
=
2(1)
1
−
2
2
=2
=
1, 1
x y
0 0
1 -1
2 0
3 3
− ʃ ʃ − − − − − − =
2
=
2
4
=
5
=
5
32
=
5
96
=
+
4
2
3
4
4 4
5
1
=
2
2
5
+4
-1 dx
2
2
y (1,-1)
-2
2
4 3 3
24 +
16 +
1
2
4 3
2
3
0
32 3
240 + 160 15
16
=
15
=
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65
EXERCISE 12.4
VOLUME OF A SOLID OF REVOLUTION
− − − − − − − − − .
+
=5;
=0;
=0 ;
=5
=0;
=5
2
=
2
=0;
;
= 5
2
=
2
5
=0
=5
5
=
25
0
2
10 +
0
=
25
= =
10
3
+
2
25 5
125
2
5 5
125 +
3 2
+
1 3
125
0
3
=
y
=0
5
1
3
0
.
+
=6 ;
= (6
=3;
=0;
)
=
2
=
6
=
36
2
12 +
3
=
0
36
12 +
2
0
2
=
36
=
36 3
=
12
2
+
6 9 +
= [ 36 3 = (9)(12
+
2
6 3
36 3
3
3
1
3
3
2
0
1
3 27
6 9 + 9]
6+1)
= (9)(7)
=
=0
(0,6)
3 2
5
− − − − − − − − − − − +
5
y
=6
=3
3
3
5
x
=0
(6,0)
1
0
1
3
5
DIFFERENTIAL & INTEGRAL CALCULUS |
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66
EXERCISE 12.4
VOLUME OF A SOLID OF REVOLUTION
.
= 4,
= 2,
= 4;
− − − − − − − − − = = =
2
(4
4
)
2
4
2
2
=
0
(16
32
0
=
16 2
32 2
=8
4
2
=
4
+
16 ) 2
16 2
1
.
0
2
=4
= 4
,
=
;
=
− − − − − − − − − − − − 2
= =
)2
(
2
2
=
(
0
2
2
=
)2
4
2
2
(
2
2
4
+
16
3
2
=
=
6
2
=4
=
2
3
2
6
1
3
+
2 3
2
)
5
+ 2
5
16(5)
+
2
16 5
2
2
2
2
16
1
3
+
− 2
16(5)
5
.
DIFFERENTIAL & INTEGRAL CALCULUS |
Feliciano & Uy
5
67
2
EXERCISE 12.4
VOLUME OF A SOLID OF REVOLUTION
.
=
,
= 0,
= 1;
=1
− − − − − − − − − − 2
=
=
)2
(1
2
=
0
)2
(1
0
2
=
1
2
2
+
0
=
[ +2 3
=
2
3
=
2
3
= =
=
4
3 2 4
+
+2
2
2
+
2
2
4
2
+2
+0
]
4
4
4(0)
0+2+0
2
.
DIFFERENTIAL & INTEGRAL CALCULUS |
Feliciano & Uy
68
EXERCISE 12.5
1.
THE WASHER METHOD
− 2
=
,
= 3,
= 0;
− − − − 9
32
=
2
0
9
=
9
0
2
=
=
9
9
2 0
9(9)
(9)2 9 2
0
=
3.
− 2
x 0 a
=4
,
= ;
y 0 2a
2
dy
=4
x
− − − − − − 2
2
=
X=a
2
2
2 2
2
=
(
2
)
4
2
2
4
2
=
16
2
− − − − − − − 5
=
=
=
2
2
2
80
(2
3
(2
3
2
32
5
80
2
2 3 5
)
2
)
( 2
(2
3
+
3
+
2 3 5
32
5
80
2
)
)
=
DIFFERENTIAL & INTEGRAL CALCULUS |
Feliciano & Uy
69
EXERCISE 12.5
THE WASHER METHOD
− −− − − 2
5.
+
=4
2
,
=
2
2
0
3
+
a
(-a,0)
3
3
=4
+
3
2
=4
=4
2
=
(a,0)
o
3
2 3 3
x=b
=
− − − 7.
2
=
=
+
2
5 0
= 25 ,
+
2
25
=5; 5
=0
2
.
DIFFERENTIAL & INTEGRAL CALCULUS |
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70
EXERCISE 12.5
9.
THE WASHER METHOD
− 2
=4 ,
2
=4 ;
− − − − 2
=
1
2 2
4 =
4
3
64 = = 4,
= 4:
(4,4)
4
=
2 2
2
4
4
0
4
4
=
4
16
0
5
2
=
2
=
2(4)2 +
4
80 0
(4)5 80
=
11.
2
=8 ,
=2 ;
=4
2
=
1
2
8 =2
8 =4
2
= 2,
=
=
4
2
= 4: 4 3 2 3
2
4(2) +
(2,4)
0
4(2)3 3
=
DIFFERENTIAL & INTEGRAL CALCULUS |
Feliciano & Uy
71
EXERCISE 12.6
THE CYLINDRICAL SHELL METHOD
. 4
=
3
,
= 0,
= 2, ;
− − − −
V = 2π
2 0
V = 2π
2 0
3
2
4
2
V = 2π
V = 2π V = 2π V = 2π
4
2 [ 0 2 4
4
dx
4
]
5
20
2 0
(2)4
(2)5
4
20
2 0
= 2
– = 4
2
,
,
= 0
− − − − − − − − 3
V = 2π 0
3
V = 2π 0
3
V = 2π V = 2π
2
4
2
V = 2π 0 4
3 5
=
4
3
1
2
3
3
1
4
4
4
4
V = 2π (3)3
2
3
3
3 0
(3)4 4
3 0
V=
DIFFERENTIAL & INTEGRAL CALCULUS |
Feliciano & Uy
72
3 0
EXERCISE 12.6
THE CYLINDRICAL SHELL METHOD
− − =
5.
,
=
,
=
2
2
=2
4
=
+
.
2
− =2
7.
,
9 0
=2
=0 ,
9
Y=9
=0
2
=
.
9. =
,
=2
=
= ,
=0
1
.
(e,1)
.
(1,0) X=e
DIFFERENTIAL & INTEGRAL CALCULUS |
Feliciano & Uy
73
EXERCISE 12.6
11.
THE CYLINDRICAL SHELL METHOD
2
= 8 ,
=0,
= 4 ; about = 4
− − − – – − – − − − − − − − −− 4
2
=2
4
8
0
4
=
=
4
2
4
3
0
4 3
4
4
3
4
4 0
=
13. (
3 ) 2 +
2
= 9;
3
.
2
x 9 ( x 3) dx
=8
0
=8 (
( 9 ( x 3
= 8 (27
27 = 8 ( )( 2
3 ) 2 3 27 )2 + 2
27 2
1+
3
3
+
9 2
(
3)( 9
3
2
3 0
1)
)
= 108 ( ) 2
=
DIFFERENTIAL & INTEGRAL CALCULUS |
Feliciano & Uy
74
EXERCISE 12.6
THE CYLINDRICAL SHELL METHOD
− − − − − − − − − − − − − − − − 2
15.
2
+
2
=
;
=
2
=
2
= =
>
2
+
2
2
+
2
4
:
2
+
2
=
2
=
2
=4
=
2
2
ln
+
2
2
2
=4
2
2
2
2
2
+
=
2
a
a
+
2
=
2
a
a
DIFFERENTIAL & INTEGRAL CALCULUS |
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75
EXERCISE 12.7
2
.
+
2
VOLUME OF SOLIDS WITH KNOWN CROSS SECTIONS
= 36
. 9
− − − − − 2
=
,
2
=2
2
,
6
=
6
6
=
2
2
6
=
=
6 2(3 6
2
.
)
36
+ 16
2
2
= 144
− 1
=
=2 =
2
2
1
=
2
(2 )( )
2
=
8
2
=2
0
=2
=
8 144 9 2 0 16
.
DIFFERENTIAL & INTEGRAL CALCULUS |
Feliciano & Uy
76
EXERCISE 12.7
VOLUME OF SOLIDS WITH KNOWN CROSS SECTIONS
.
− − − = (1
2
)(2
)
2
=2
(1
)2
2
0
2
=2
0
=
2
(1
4
)
2
64 15
= .
.
DIFFERENTIAL & INTEGRAL CALCULUS |
Feliciano & Uy
77
EXERCISE 12.8
LENGTH OF AN ARC
.
3 2
=
=0
3 2
=
= =
3 2 3 2
1 2 1 2
5
=
)2
1+(
0
=
=
5 0
5 0
=
1+(
1+
3 2
1 2
)2 dx
9
4
.
=5
− 2 3
3.
1 3
9
=
1+
0
2 3
9
=
2 3
:
2 3
=
2 3
2
1 3
1 3
+
2 3
2 3
9
=
2 3
0
1 3
9
=
1 3
0
=
+
2 3
0
3
1 3
=
+
2 3
+
2
2 3
9 0
3
2
=4
3
2
=
X=0
x=5
DIFFERENTIAL & INTEGRAL CALCULUS |
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78
EXERCISE 12.8
LENGTH OF AN ARC
=
,
=
;
1
=
6
=
2
=
=
2
1+
= ( = (
)
= (1 = (
)
)
2
=
2
1
+
2
=
2 0
2
1
sin2
+ sin2
=
2
6
= .
− − − − − − − =2 1
9.
2
=2 1 =2 =2
2
)2
= 4(1 2
=
4(1 4(1
)2 + 4
2
0
6
=2
2 0
(1 (1
)2 +
2
=
DIFFERENTIAL & INTEGRAL CALCULUS |
Feliciano & Uy
,
) )
0
6
=
)
=
2
1+
= (1
2
=
2
=
= (1
=
=
− − − −− − − − − − −
79
EXERCISE 12.9
AREA OF A SURFACE OF REVOLUTION
− − − − − − − −− − − − 2
2
+
= 16 ;
=2
4
=2
2
=
=
=
2
16
1 2
1 2
2
16
( 2 )
2
16
2
=
1+
2
=
=
=
1+
2
16
2
16
+
2
2
16
4
2
16
4
=2
16
2
=2 =
4 4 2
.
2
4
16
2
=4
2
= 12
;
=0
=3
3
=2
0
= 12 1
=
2
(12)
6
=
12
=
1+
36
12
12 + 36
=
=
1 2
12
12
2 3 +9 12 3
=2
2 3 +9
12
12
0
=4
3 0
=
.
=
3
=3
2
=
=2
= .
3 +9 .
;
=0
1+9 1 0
3
=1
4
1+9
4
.
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EXERCISE 12.9
AREA OF A SURFACE OF REVOLUTION
− =
2
;
=0
=
4
4
=2
0
=
=
2 (2) 2
1+4
=2
4
0
2
2
= .
1+4
22
.
− − 2
4
=0
=2
2
=2
0
=
2
= 1 + 4 ^2 2 0
=2 =
.
1+4
2
.
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EXERCISE 12.9
AREA OF A SURFACE OF REVOLUTION
− 13.
=
;
=0;
=1;
1
= 2
1 +
2
0
1
= 2
1 +
2=
0
= 2 = 2 =
1 +
1 +
2( 2/2 )
1 0
2( ½ )
+
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EXERCISE 13.1
FORCE OF FLUID PRESSURE
= = (62.5 / = 24000
1.
3
2
)(96
)(4
=
=
3.
)
1
=
2
5 3
2 2 + ( )(3) 3
=
=
2
=
=( =
=
3
62.5
)(4
1
)( 144
2
(625)(4)
5
144
.
= 50
5.
12ft
8ft
3
) 2
=3
=
50 =
1
3
2
1 3
2
50 =
2
100 =
2
=
3
h
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EXERCISE 13.1
FORCE OF FLUID PRESSURE
7.
=
=
[( )(3)(2)](2)
=
= 6 = major axis = 4 =
0
y
b a
A=
x
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EXERCISE 13.2
WORK
1.
− − =
=
40
=
;
1
=
2
=
,
1
,
2
= 40
;
= 0,
= 14
10=4
= 80
4 80 0
=
3.
=
=
=
;
50
−
=
1 10
,
= 5
= 0,
=
0
=
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EXERCISE 13.2
WORK
− − − − − − − 5.
=
=
60
2
=
=9
60
(60
)
10
=9
0
60
0
=9
60
=9
60
=9
600
2
10 0
2
=
10 2 0
50 .
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EXERCISE 13.2
WORK
9.
=
− − − − − − − =
2
+
2
=
;
2
;
2
=
2 ;
= 10
,
=2 ,
=
=2
2
=
6
10
2
2
2
= 20
6
22
2
2
=
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EXERCISE 13.3
FIRST MOMENT OF A PLANE AREA
− 2
1.
=4 ,
=
=4
2
1
4 4 2 0
2
=4
=4
=
=
4 0
=
4
.
2
− − − 4
4 =
4
64
=4
= 0,
2
1
4
=
=4
2
=0
=4
4
4
16
=
− − −
=
=0
3
64
1
16
4
=
4
4
=
2
4
4
=
=
2
4
16
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EXERCISE 13.3
FIRST MOMENT OF A PLANE AREA
−− − −− − −− −− − −−−− − − − − 3 0
=
7.
3
=
0
=3
3
= 27
3
2
3
2
9
)2
(3 0
2
=
2
9
= ;
3
=
=
= 0;
=0
2
3
3
= 3;
=
3
2
− − −− − − = = =
27
4
27
=
.
9
4 27
9
18
72
4
[
]
2
=4
,
=
0
2
2
1+
2
= 27
0
= 27
)2
(3
3
3
)2
(3
1
0
9
=
=3
3
3
2
9
3
)2 (3 + )2
3 9
3 0
=
2
9
3+
3 (3 0
3
=
[
3
0
=
*
3
2
0
2
+
2
2
= 27
=
27
− − −− −− − − − − − − −− − − −− 2
4
0
4
4
3
*
=
2
9
0
2
=9 = 2
= = =
*
1
| 30
3 2
2
2
=
2
9
1 2
( )[(9
2 3 1 3
9)
27 =
3 (3 0
=
@
3 2
9
)2
(9
3 2
0) ]
= 3; = 0;
=0 =9
= = =
− − −
1
2
2
2
3 2 0 1
4
2 2
2
=3 =
= = =
)3
(3
3
0
33
3
3
| 30
9
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EXERCISE 13.4
CENTROID OF A PLANE AREA
− − − − − − + 2 = 6,
= 0,
=0
Solving for A
=
6 0
=
6 0
3
2
= [3
4
] 60
36
= 3 6 =
2
4
.
− − − −
Solving for
=
6 0
=
6 0
=
6 0
=[
Solving for
3 2 2
=
2
3
=
2
2
3
=
2
3
3
] 60
6 0
6 (3 0 2 1
2
2
)
2
6 (9 0 2 1
3 +
1
3
2
2
= [9
) (3
2
4
3
+
12
)
] 60
1
9 = 18
= (3) 3
=
=
Centroid: (2,1)
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EXERCISE 13.4
3.
CENTROID OF A PLANE AREA
− =
,
=0
=0
− − − − − − − −
A= = 0 =
− − − =
= = = =
;
=
( )
0
2
1
2
2 0 1
2
2 0 1
( 0
2
1
1
= (
2
1
2
2
= (
2 2
=
2
4
4
)
2
2
2 2
=
2
(2)
)
2
A=2
=
= =
;
0
=
0
0
=
;
=
=
;
=
=
= [
+
=
+
]
+0
0
=
= ( )(2) 4
=
Centroid:
8
,
DIFFERENTIAL & INTEGRAL CALCULUS |
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EXERCISE 13.4
CENTROID OF A PLANE AREA
2
3
=
,
=2
− − − − − − 4
=
3 2
(2
)
0
=[
2
2
5
64
= [16
=
− − − − − 4
=
=
3 2
(2
)
=
(2
5 2
2
)
=
=
=
2
3
3
2 7
5 2 [ (4)3 16 3
5 128 [ 16 3
7 2
]
2 7
257 ] 7
.
5
=
0
=[
]
=
0
4
16
5
]
=
7
(4)2
=
4
1
=
0
4
5 2
2
1 2 1 2
2
0
4
[(2 )
5 2
2
]
0
4
(4
2
3
)
0
1 4 2 3
4
3
4
4 0
10 3
40 21
:
,
DIFFERENTIAL & INTEGRAL CALCULUS |
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EXERCISE 13.4 9.
CENTROID OF A PLANE AREA
2
2
+
= 25,
+
=5
− − − − − − − − − − − −− −− − − − − − − − − − − − − − − − − − − ∴ − ∙ − − − − − − 25
=
50
4
25
=
(
4
2)
5
=
5
=
−−−− − ∶ − − ∙ → ∏/2 ∶−− − ∶ ∴ − − 5
=
2
25
0 5 0
=
2
25
A 25
5 0
=
+
C
2
=
=
=
2
25
;
@ = 5;
=0;
5 0
B
5sin
2
5 0
5
5cos
5cos
25
5
= arcsin =
5
2
=0
5cos
5cos
0
2
= 25
cos 2
cos 2
0
1
= 25
2
= 25 =
2
0
4
5 25
cos2
0
0
5
= =
2
2
+0
4
25
+
1
5 0
5 0
2 5
=
25
2 0
2
=
25
= 25
4
25 +
25 2
25 2
=
1+
2
2
2
=
2
+ 125
2
+
2 250
3
6
0
5 0
3
125
0+0
3 375+ 500
2
2
5
2
5
2
0
2
5
1
2
25
0
3
1
25
2
1
=
1/2 25 +
125
1
3
2
125
125 +
6
=
6
25
3
125
5
0
2
=
125 6
4
25
2
4
10
3
125
3
:
125
=
2
3
3
125
2
=
3
0
5
1
5
6
25
2
+
5
1
0
3
3
+ = 2 3 125
=
2
5
125
3 125
=
=
3
0
=
2
3 2 2
25
=
=
3 2
2
=
5 2
2
+
0
3 2 2
1 25
=
5
2
= 25 = 2
5
2
25
0
x
5
0
5
5
2
25
2
125
= =
=
2
4
10
3
6
25
2
DIFFERENTIAL & INTEGRAL CALCULUS |
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EXERCISE 13.5
CENTROID OF A SOLID OF REVOLUTION
− − − −− − − − − − − 2
1.
=
;
=3;
=0;
.
2
3
= 4,
= 1,
= 4,
=
2
=
4
=2
=
2
1
4
4
2
2
2
2 4
= 16
3
2
1
0 9
0 3
= 2
2
1
1
2
2
0
9
=
= 2
0
= 381.70
4
3
0
4
=
=
2
1
3+
4
2
4
= 16
152.68
=
3
=
1
4
16 3
3
1
21
4
= 2.5 , . ,
2
1
3 4
4
= 16
=
= 16
4
1
2
381.70
4
16
1
=
=
4
9
=2
=
8
15
=
9
4
1
= 16
=
2
1
= 16
4
4
=
2
4
=
1
1
;
4
4
16
=
=
=0
=
15 2 21 4
=
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EXERCISE 13.5
CENTROID OF A SOLID OF REVOLUTION
− .
2
x 0 1 2 4
=4
2
,
x 0 1/4 1 4
y 0 ¼ 1 4
=4
y 0 1 2 4
− − − 4
=
2 2
2
4
4
0
4
4
=
4
16
0
=
96
5
4
=2
0
=
2
4 + 2
4
2
4
4
128 3
=
=
=
,
128 3 96 5
,
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EXERCISE 13.5
CENTROID OF A SOLID OF REVOLUTION
.
2
=4 ,
X 0 1/4
Y 0 1
1 4
2 4
=
X 1 2 3 4
=0
Y 1 2 3 4
− − 4
=2
( 4
)
0
= 26.80829731
.
=2
4
=2
(
0
4 + 2
)
4
= 64 /3
=
= 2.5
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EXERCISE 13.6
MOMENT OF INERTIA OF A PLANE AREA
−
1. 2 +
=6,
=0,
=0;
5.
=2
,
= 0,
dy
x
y
4-y
=4
(4,4)
dx
− − − − − − 0 3
6 0
=
6 0
2
6 0
=
2 1
=
2 1
=
2
0 4
2
6
1
=
6
2
2
6
3
4
0
0
3
4
2 6
6
3
4
4
=
4
=
3
=
,
=8,
2
4
2
4
0
=
3. 0
2
=
4
2
0 4
=0;
=
4 0
2
2
=
dy x
= =
2 2 0 2 5 0
=
=
=
2 2 ( 3) 0 6
6
=
26 6
DIFFERENTIAL & INTEGRAL CALCULUS |
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EXERCISE 13.6
7.
MOMENT OF INERTIA OF A PLANE AREA
2
=8
,
=2
9.
− 2
=4
,
= 4 ;
=4
2
=4
(1,4)
X1 X2
dy
dx
y (0,0)
− − − 0 1 2
0
2 2 4
0 0 1 2 2 4
4
2
=
(
)
0
4
0
=
0 1
0 4
0 1
0 4
− − 2
= =
1 0
2
(
(4
)
4
2
)
2
2
=
4 3 ( 0 2
(
2
8
)
4
8
)
=
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EXERCISE 13.6
11.
MOMENT OF INERTIA OF A PLANE AREA
2
= 8 ,
=0,
= 4 , with respect to
=4
−−−
13.
=
,
=2
,
+
= 6,
+
(6
=6
2 )
6
3
=2
− − − 4
2
=
2
4
8
0
=
=
=
1 8
4
16
8
3
+
0
1 16 3 8
2
3
2
4
5
+
4
5 0
4
− −− 0 1 2
0 1 2
0 0 1 2 2 4
0 0 1 5 2 4
=
=
=
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=0
EXERCISE 13.7
MOMENT OF INERTIA OF A SOLID OF REVOLUTION
=2
,
=0,
− − 4
3
=2
2
0
0
4
7
=4
2
0
=4
=4
=4 =
9
2
4
2
9
2 4
9
1024
0
9
2
2 0
9
+
= 4 ;about = 0
9
4
2
0
=
,
=0,
= 0 ;about the y-
axis
− − − − − 3
=2
0
0
=
2
3
4
0
=
2
3
4
0
=
=
9
=
=
0
− 4
2
4
5
5
4
5
2
2
0
5
5
09
5
20
DIFFERENTIAL & INTEGRAL CALCULUS |
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EXERCISE 13.7
MOMENT OF INERTIA OF A SOLID OF REVOLUTION
2 +3 =6, axis
=0,
− − − − 3
=
=
2
6
2
0
3 16 4 192 3 +864 2 1728 +1296 2 0 81
=
2
= 3 ,
=4,
= ,
= 1 ; about = 0
4
3
0
= 0 ; about the x-
= ; about = 0
− − − − 2
3
=2
4
1
2
=2
2
4
4
1
3 2
=2
=2
4
3
1
28
31
3
5
5 2
5
1
=
− − − − 3
3
=2
3
0
3
7
=2
3
2
4
0
3
=2
7
3
0
=2
54
3
4
2
0
243 5
=
DIFFERENTIAL & INTEGRAL CALCULUS |
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EXERCISE 13.7
MOMENT OF INERTIA OF A SOLID OF REVOLUTION
2
=
,
= 2 ;about the y-axis
− − − − −
3
=
,
=1,
= 0 ; about
− =
1
2
3
=2
2
2
0
2
=2
4
2
5
0
2
=2
2
4
2
5
0
0
5 2
=2
=2
2
5
0
64
32
5
3
6 2
6
0
− 1
=2
+1
3
3
0
0
1
6
=2
+3
5
4
+3
+
3
0
7
=2
7
6
+
2
+
3 5 5
4
+
1
4 0
=
=
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